Showing papers on "Fourier series published in 2012"
TL;DR: In this paper, a unit-root test based on a simple variant of Gallant's (1981) flexible Fourier form is proposed. But the test relies on the fact that a series with several smooth structural breaks can often be approximated using the low frequency components of a Fourier expansion, thus it is possible to test for a unit root without having to model the precise form of the break.
Abstract: We develop a unit-root test based on a simple variant of Gallant's (1981) flexible Fourier form. The test relies on the fact that a series with several smooth structural breaks can often be approximated using the low frequency components of a Fourier expansion. Hence, it is possible to test for a unit root without having to model the precise form of the break. Our unit-root test employing Fourier approximation has good size and power for the types of breaks often used in economic analysis. The appropriate use of the test is illustrated using several interest rate spreads.
514 citations
Posted Content•
TL;DR: In this paper, the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal was considered, and an O(k log n log(n/k))-time randomized algorithm for general input signals was proposed.
Abstract: We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show:
* An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and
* An O(k log n log(n/k))-time randomized algorithm for general input signals.
Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k = o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = n^{\Omega(1)}.
We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least \Omega(k log(n/k)/ log log n) signal samples, even if it is allowed to perform adaptive sampling.
260 citations
Book•
12 Jun 2012
TL;DR: The Real Number System (RNS) as discussed by the authors is a generalization of the metric space theory of functions on R. The Riemann-Stieltjes Integral Integral and Functions of Bounded Variation.
Abstract: The Real Number System.- Continuity and Limits.- Basic Properties of Functions on R.- Elementary Theory of Differentiation.- Elementary Theory of Integration.- Elementary Theory of Metric Spaces.- Differentiation in R.- Integration in R.- Infinite Sequences and Infinite Series.- Fourier Series.- Functions Defined by Integrals.-Improper Integrals.- The Riemann-Stieltjes Integral and Functions of Bounded Variation.- Contraction Mappings, Newton's Method, and Differential Equations.- Implicit Function Theorems and Lagrange Multipliers.- Functions on Metric Spaces.- Approximation.- Vector Field Theory the Theorems of Green and Stokes. Appendices.
158 citations
TL;DR: Theorem 1.1 is intimately related to almost everywhere convergence of partial Fourier sums for functions in L[0, 1] as mentioned in this paper, and it is indeed equivalent to the celebrated theorem by Carleson [2] for p = 2 and the extension of Carlesone's theorem by Hunt [9] for 1 < p < ∞.
Abstract: By a standard approximation argument it follows that S[f ] may be meaningfully defined as a continuous function in ξ for almost every x whenever f ∈ L and the a priori bound of the theorem continues to hold for such functions. Theorem 1.1 is intimately related to almost everywhere convergence of partial Fourier sums for functions in L[0, 1]. Via a transference principle [12], it is indeed equivalent to the celebrated theorem by Carleson [2] for p = 2 and the extension of Carleson’s theorem by Hunt [9] for 1 < p < ∞; see also [7],[15], and [8]. The main purpose of this paper is to sharpen Theorem 1.1 towards control of the variation norm in the parameter ξ. Thus we consider mixed L and V r norms of the type:
125 citations
TL;DR: A Fourier regression algorithm is illustrated, here on time series of normalized difference vegetation indices (NDVIs) for Landsat pixels with 30-m resolution, which indicates that Fourier regressors may be used to interpolate missing data for multitemporal analysis at the Landsat scale, especially for annual or longer studies.
Abstract: With the advent of free Landsat data stretching back decades, there has been a surge of interest in utilizing remotely sensed data in multitemporal analysis for estimation of biophysical parameters Such analysis is confounded by cloud cover and other image-specific problems, which result in missing data at various aperiodic times of the year While there is a wealth of information contained in remotely sensed time series, the analysis of such time series is severely limited due to the missing data This paper illustrates a technique which can greatly expand the possibilities of such analyses, a Fourier regression algorithm, here on time series of normalized difference vegetation indices (NDVIs) for Landsat pixels with 30-m resolution It compares the results with those using the spatial and temporal adaptive reflectance fusion model (STAR-FM), a popular approach that depends on having MODIS pixels with resolutions of 250 m or coarser STAR-FM uses changes in the MODIS pixels as a template for predicting changes in the Landsat pixels Fourier regression had an R2 of at least 90% over three quarters of all pixels, and it had the highest RPredicted2 values (compared to STAR-FM) on two thirds of the pixels The typical root-mean-square error for Fourier regression fitting was about 005 for NDVI, ranging from 0 to 1 This indicates that Fourier regression may be used to interpolate missing data for multitemporal analysis at the Landsat scale, especially for annual or longer studies
118 citations
TL;DR: A simple framework to derive and analyse a class of one-step methods that may be conceived as a generalization of the class of Gauss methods, from which a large subclass of Runge–Kutta methods can be derived.
114 citations
TL;DR: The developed solver is shown to provide high order solutions, second order in time convergence rates and spectral convergence when solving the incompressible Navier-Stokes equations on meshes where fixed and rotating elements coexist.
94 citations
Book•
21 Oct 2012
TL;DR: In this article, a Riemannian space associated to L 1(?n) is defined, and the heat kernel for the sublaplacian is estimated for the Weyl transform.
Abstract: 1 Euclidean Spaces.- 1.1 Fourier transform on L1(?n).- 1.2 Hermite functions and L2 theory.- 1.3 Spherical harmonics and symmetry properties.- 1.4 Hardy's theorem on ?n.- 1.5 Beurling's theorem and its consequences.- 1.6 Further results and open problems.- 2 Heisenberg Groups.- 2.1 Heisenberg group and its representations.- 2.2 Fourier transform on Hn.- 2.3 Special Hermite functions.- 2.4 Fourier transform of radial functions.- 2.5 Unitary group and spherical harmonics.- 2.6 Spherical harmonics and the Weyl transform.- 2.7 Weyl correspondence of polynomials.- 2.8 Heat kernel for the sublaplacian.- 2.9 Hardy's theorem for the Heisenberg group.- 2.10 Further results and open problems.- 3 Symmetric Spaces of Rank 1.- 3.1 A Riemannian space associated to Hn.- 3.2 The algebra of radial functions on S.- 3.3 Spherical Fourier transform.- 3.4 Helgason Fourier transform.- 3.5 Hecke-Bochner formula for the Helgason Fourier transform.- 3.6 Jacobi transforms.- 3.7 Estimating the heat kernel.- 3.8 Hardy's theorem for the Helgason Fourier transform.- 3.9 Further results and open problems.
90 citations
TL;DR: In this article, the free vibration and dynamic response of simply supported functionally graded piezoelectric cylindrical panels impacted by time-dependent blast pulses are analyzed using Hamilton's principle, the equations of motion based on first-order shear deformation theory are derived.
86 citations
TL;DR: In this paper, the Fourier series expansion method was applied to investigate the vibration characteristics of thin rotating cylindrical shells under various boundary conditions, and the results are presented for a thin rotating cylinder with classical boundary conditions of any type.
84 citations
TL;DR: In this paper, a simple and efficient method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis, is introduced.
TL;DR: In this article, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed, and certain applications are also given, such as this article.
Abstract: In this survey, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed. Certain applications are also given.
TL;DR: This paper builds upon existing shapebased techniques to present an alternative Fourier series approximation for rapid low-thrust-rendezvous/orbitraising trajectory construction with thrust-acceleration constraint-handling capability, and shows its ability to solve problems with a greater number of free parameters than in shape-based methods.
Abstract: Space mission trajectory design using low-thrust capabilities is becoming increasingly popular. However, trajectory optimization is a very challenging and time-consuming task. In this paper, we build upon existing shapebased techniques to present an alternative Fourier series approximation for rapid low-thrust-rendezvous/orbitraising trajectory construction with thrust-acceleration constraint-handling capability. The new flexible representation, along with the constraint-handling capability, makes this method a suitable candidate for feasibility assessment of a whole range of trajectories within the given system propulsive budget. In addition, the solutions present opportunities for direct optimization techniques. A key point in the proposed method is its ability to solve problemswith a greater number of free parameters than in shape-basedmethods. Several case studies are presented: simpleEarth–Mars rendezvous, rendezvous/orbit raising for lowEarth orbit to geostationary orbit, and two phasing problems. Results clearly depict the advantage of the proposed method in handling thrust constraints and its applicability to a wide range of problems.
TL;DR: In this paper, a generalized warblet transform (GWT) is proposed to characterize highly oscillating time-frequency patterns of signals, whose instantaneous frequency (IF) is periodic or non-periodic.
TL;DR: In this article, the wave equation in fractal vibrating string was introduced in the framework of local fractional calculus and the technique of the LFTF series was applied to derive the solution of the Local fractional wave equation.
Abstract: We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the MittagLeffler function.
01 Mar 2012
TL;DR: In this paper, an upper bound for the size of the first sign-change of Hecke eigenvalues in terms of conductor and weight was given, and the authors investigated to what extent the signs of Fourier coefficients determine an unique modular form.
Abstract: We consider two problems concerning signs of Fourier coefficients of classical modular forms, or equivalently Hecke eigenvalues: first, we give an upper bound for the size of the first sign-change of Hecke eigenvalues in terms of conductor and weight; second, we investigate to what extent the signs of Fourier coefficients determine an unique modular form. In both cases we improve recent results of Kowalski, Lau, Soundararajan and Wu. A part of the paper is also devoted to generalized rearrangement inequalities which are utilized in an alternative treatment of the second question.
Book•
06 Jul 2012
TL;DR: In this article, Fourier Transforms of Stable Signals, Fourier series of Locally Stable Periodic Signals and Pointwise Convergence of Fourier Series are discussed.
Abstract: A. FOURIER ANALYSIS IN L1 Fourier Transforms of Stable Signals / Fourier Series of Locally Stable Periodic Signals / Pointwise Convergence of Fourier Series B. SIGNAL PROCESSING Filtering / Sampling / Digital Signal Processing / Subband Coding C. FOURIER ANALYSIS IN L2 Hilbert Spaces / Complete Orthonormal Systems / Fourier Transforms of Finite Energy Signals / Fourier Series of Finite Power Periodic Signals D. WAVELET ANALYSIS The Windowed Fourier Transform / The Wavelet Transform / Wavelet Orthonormal Expansions / Construction of a MRA / Smooth Multiresolution Analysis
TL;DR: In this paper, the vibration behaviors of a box-type built-up structure and energy transmission through the structure are investigated analytically, and the results of the power flow and structural intensity are presented to obtain a clear physical understanding of the physical mechanisms of energy transmission.
TL;DR: In this paper, a new fully numerical method is presented which employs multiple Poincare sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP), which reduces the calculations required for searching two-dimensional invariant tori to a search for closed orbits.
Abstract: A new fully numerical method is presented which employs multiple Poincare sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory required to store these orbits. This method reduces the calculations required for searching two-dimensional invariant tori to a search for closed orbits, which are the intersection of the invariant tori with the Poincare sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant tori is reduced to maps between the consecutive Poincare maps. A Newton iteration scheme utilizes the invariance of the circles of the maps onthesePoincaresectionsinordertofindtheFouriercoefficientsthatdefinethecirclestoany given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasiperiodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-halo and Lissajous families of the Sun-Earth RTBP around the L2 libration point are obtained via this method. Results are compared with the existing literature. A numerical method to transform these orbits from the RTBP model to the real ephemeris model of the Solar System is introduced and applied.
TL;DR: In this article, the shape variation in the shape of cereal grains, namely; barley, oat, rye and wheat (Canada Western Amber Durum and Canada Western Red Spring), were quantitatively evaluated using principal components analysis (PCA) based on elliptic Fourier descriptors.
01 Jan 2012
TL;DR: In this article, the authors present a self-contained presentation of the theory that has been developed recently for the numerical analysis of the controllability properties of wave propagation phenomena and, in particular, for the constant coefficient wave equation.
Abstract: In these Notes we make a self-contained presentation of the theory that has been developed recently for the numerical analysis of the controllability properties of wave propagation phenomena and, in particular, for the constant coefficient wave equation. We develop the so-called discrete approach. In other words, we analyze to which extent the semidiscrete or fully discrete dynamics arising when discretizing the wave equation by means of the most classical scheme of numerical analysis, shear the property of being controllable, uniformly with respect to the mesh-size parameters and if the corresponding controls converge to the continuous ones as the mesh-size tends to zero. We focus mainly on finite-difference approximation schemes for the one-dimensional constant coefficient wave equation. Using the well known equivalence of the control problem with the observation one, we analyze carefully the second one, which consists in determining the total energy of solutions out of partial measurements. We show how spectral analysis and the theory of non-harmonic Fourier series allows, first, to show that high frequency wave packets may behave in a pathological manner and, second, to design efficient filtering mechanisms. We also develop the multiplier approach that allows to provide energy identities relating the total energy of solutions and the energy concentrated on the boundary. These observability properties obtained after filtering, by duality, allow to build controls that, normally, do not control the full dynamics of the system but rather guarantee a relaxed controllability property. Despite of this they converge to the continuous ones. We also present a minor variant of the classical Hilbert Uniqueness Method allowing to build smooth controls for smooth data. This result plays a key role in the proof of the convergence rates of the discrete controls towards the continuous ones. These results are illustrated by means of several numerical experiments.
TL;DR: In this paper, a new theory of two-temperature generalized thermoelasticity is constructed in the context of a new consideration of heat conduction with fractional orders.
Abstract: In this paper, a new theory of two-temperature generalized thermoelasticity is constructed in the context of a new consideration of heat conduction with fractional orders. The two-temperature Lord–Shulman (2TLS) model and two-temperature Green–Naghdi (2TGN) models of thermoelasticity are combined into a unified formulation using the unified parameters. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain which is then solved by using a state-space approach. The inversions of Laplace transforms are computed numerically using the method of Fourier series expansion technique. The numerical estimates of the quantities of physical interest are obtained and depicted graphically. Some comparisons of the thermophysical quantities are shown in figures to estimate the effects of the temperature discrepancy and the fractional order parameter.
TL;DR: In this article, the authors investigated the effects of the presence of a transverse crack in a rotating shaft under uncertain physical parameters in order to obtain some indications that might be useful in detecting the presence a crack in rotating system.
TL;DR: In this paper, an exact analytical solution of transient heat conduction in cylindrical multilayer composite laminates is presented, which is valid for the most generalized linear boundary conditions consisting of the conduction, convection and radiation heat transfer.
TL;DR: In this article, an analytical solution for a plate made of functionally graded materials based on the third-order shear deformation theory and subjected to lateral thermal shock is presented for the analytical solution is obtained under coupled thermoelasticity assumptions.
Abstract: In this paper, the analytical solution is presented for a plate made of functionally graded materials based on the third-order shear deformation theory and subjected to lateral thermal shock. The material properties of the plate, except Poisson's ratio, are assumed to be graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The solution is obtained under the coupled thermoelasticity assumptions. The temperature profile across the plate thickness is approximated by a third-order polynomial in terms of the variable z with four unknown multiplier functions of ( x , y , t ) to be calculated. The equations of motion and the conventional coupled energy equation are simultaneously solved to obtain the displacement components and the temperature distribution in the plate. The governing partial differential equations are solved using the double Fourier series expansion. Using the Laplace transform, the unknown variables are obtained in the Laplace domain. Applying the analytical Laplace inverse method, the solution in the time domain is derived. Results are presented for different power law indices and the coupling coefficients for a plate with simply supported boundary conditions. The results are validated based on the known data for thermomechanical responses of a functionally graded plate reported in the literature.
TL;DR: In this paper, the authors studied the two-temperature generalized thermoelasticity theory (2TT) for the three-phase-lag (TPL) heat equation in an isotropic infinite elastic body with a spherical cavity.
Abstract: The thermoelastic interaction for the three-phase-lag (TPL) heat equation in an isotropic infinite elastic body with a spherical cavity is studied by two-temperature generalized thermoelasticity theory (2TT). The heat conduction equation in the theory of TPL is a hyperbolic partial differential equation with a fourth-order derivative with respect to time. The medium is assumed to be initially quiescent. By the Laplace transformation, the fundamental equations are expressed in the form of a vector-matrix differential equation, which is solved by a state-space approach. The general solution obtained is applied to a specific problem, when the boundary of the cavity is subjected to the thermal loading (the thermal shock and the ramp-type heating) and the mechanical loading. The inversion of the Laplace transform is carried out by the Fourier series expansion techniques. The numerical values of the physical quantity are computed for the copper like material. Significant dissimilarities between two models (the two-temperature Green-Naghdi theory with energy dissipation (2TGN-III) and two-temperature TPL model (2T3phase)) are shown graphically. The effects of two-temperature and ramping parameters are also studied.
TL;DR: In this article, the authors show that the locations of the jumps and the pointwise values of a piecewise-smooth function can be reconstructed with at least half the classical accuracy.
Abstract: Accurate reconstruction of piecewise-smooth functions from a finite number of Fourier coefficients is an important problem in various applications. The inherent inaccuracy, in particular the Gibbs phenomenon, is being intensively investigated during the last decades. Several nonlinear reconstruction methods have been proposed, and it is by now well-established that the "classical" convergence order can be completely restored up to the discontinuities. Still, the maximal accuracy of determining the positions of these discontinuities remains an open question. In this paper we prove that the locations of the jumps (and subsequently the pointwise values of the function) can be reconstructed with at least "half the classical accuracy". In particular, we develop a constructive approximation procedure which, given the first $k$ Fourier coefficients of a piecewise-$C^{2d+1}$ function, recovers the locations of the jumps with accuracy $\sim k^{-(d+2)}$, and the values of the function between the jumps with accuracy $\sim k^{-(d+1)}$ (similar estimates are obtained for the associated jump magnitudes). A key ingredient of the algorithm is to start with the case of a single discontinuity, where a modified version of one of the existing algebraic methods (due to K.Eckhoff) may be applied. It turns out that the additional orders of smoothness produce a highly correlated error terms in the Fourier coefficients, which eventually cancel out in the corresponding algebraic equations. To handle more than one jump, we propose to apply a localization procedure via a convolution in the Fourier domain.
TL;DR: The goal of this article is to model cognitive control related activation among predefined regions of interest (ROIs) of the human brain while properly adjusting for the underlying spatio-temporal correlations while properly estimating the correlation structure in the network.
Abstract: The goal of this article is to model cognitive control related activation among predefined regions of interest (ROIs) of the human brain while properly adjusting for the underlying spatio-temporal correlations. Standard approaches to fMRI analysis do not simultaneously take into account both the spatial and temporal correlations that are prevalent in fMRI data. This is primarily due to the computational complexity of estimating the spatio-temporal covariance matrix. More specifically, they do not take into account multiscale spatial correlation (between-ROIs and within-ROI). To address these limitations, we propose a spatio-spectral mixed-effects model. Working in the spectral domain simplifies the temporal covariance structure because the Fourier coefficients are approximately uncorrelated across frequencies. Additionally, by incorporating voxel-specific and ROI-specific random effects, the model is able to capture the multiscale spatial covariance structure: distance-dependent local correlation (within ...
TL;DR: A closed-loop identification procedure using modified Fourier series as exciting trajectories satisfying the boundary conditions is proposed and implemented on the first three axes of the QIANJIANG-I 6-DOF robot manipulator.
Abstract: This paper concerns the problem of dynamic parameter identification of robot manipulators and proposes a closed-loop identification procedure using modified Fourier series (MFS) as exciting trajectories. First, a static continuous friction model is involved to model joint friction for realizable friction compensation in controller design. Second, MFS satisfying the boundary conditions are firstly designed as periodic exciting trajectories. To minimize the sensitivity to measurement noise, the coefficients of MFS are optimized according to the condition number criterion. Moreover, to obtain accurate parameter estimates, the maximum likelihood estimation (MLE) method considering the influence of measurement noise is adopted. The proposed identification procedure has been implemented on the first three axes of the QIANJIANG-I 6-DOF robot manipulator. Experiment results verify the effectiveness of the proposed approach, and comparison between identification using MFS and that using finite Fourier series (FFS)...